Brooks’ theorem for Bernoulli shifts

If Γ is an infinite group with finite symmetric generating set S, we consider the graph G(Γ, S) on [0, 1]Γ by relating two distinct points if an element of s sends one to the other via the shift action. We show that, aside from the cases Γ = Z and Γ = (Z/2Z) ∗ (Z/2Z), G(Γ, S) satisfies a measure-theoretic version of Brooks’ theorem: there is a G(Γ, S)-invariant conull Borel set B ⊆ [0, 1]Γ and a Borel coloring c : B → d of G(Γ, S) B, where d = |S| is the degree of G(Γ, S). As a corollary we obtain a translation-invariant random d-coloring of the Cayley graph Cay(Γ, S) which is a factor of IID, addressing a question from [9].